Fractal Green Function Theory

Abstract

This paper provides a comprehensive study of fractal calculus and its application to differential equations within fractal spaces. It begins with a review of fractal calculus, covering fundamental definitions and measures related to fractal sets. The necessary preliminaries for understanding fractal Green’s functions are introduced, laying the groundwork for further exploration. We develop the fractal Green’s function for inhomogeneous fractal differential equations and extend this to the fractal Helmholtz equation. The application of the fractal Green’s function to the Schrödinger equation is also investigated, focusing on the fractal Schrödinger-type differential equation with a fractal mesonic potential. Additionally, the scattering amplitude is derived within the fractal Born approximation, offering insights into scattering phenomena in fractal spaces. The findings highlight the significant impact of fractal geometry on classical and quantum mechanics and present new methods for addressing problems in fractal environments. © 2025 Elsevier B.V.